The integral


\small \int \sqrt{1 + 2cot x ( cosec x + cot x) dx}

\small \left ( 0< x< \frac{\pi }{2} \right ) is equal to

(where C is a constant of integration)

 

  • Option 1)

    4\log \left ( \sin \frac{x}{2} \right ) + C

  • Option 2)

    2\log \left ( \sin \frac{x}{2} \right ) + C

  • Option 3)

    2\log \left ( \cos \frac{X}{2} \right ) +C

  • Option 4)

    4\log \left ( \cos \frac{x}{2} \right ) +C

 

Answers (1)

As we learnt in

Integrals for Trigonometric functions -

\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos x \right ) =sinx

\therefore \int sinxdx=-cosx+c

-

 

 \int \sqrt{\left ( 1+2\cot x\ cosecx\ \ +2\cot ^{2}x \right )dx}

=\int \sqrt{\left ( 1+\cot ^{2}x \right )+2\cot x\: cosecx+\cot ^{2}x}\:dx

=\int \sqrt{\left ( cosec^{2}x+2\cot x\:cosecx+\cot ^{2}x \right )}dx

=\int \left ( cosecx+\cot x \right )dx

=ln\ tan\frac{x}{2}+ln\ sin x

=ln\frac{sin\frac{x}{2}}{cos\frac{x}{2}}\times 2sin\frac{x}{2}cos\frac{x}{2}

=ln2\sin ^2\frac{x}{2}+C

=2\ ln\frac{\sin x}{2}+C'

 


Option 1)

4\log \left ( \sin \frac{x}{2} \right ) + C

Incorrect option

Option 2)

2\log \left ( \sin \frac{x}{2} \right ) + C

Correct option

Option 3)

2\log \left ( \cos \frac{X}{2} \right ) +C

Incorrect option

Option 4)

4\log \left ( \cos \frac{x}{2} \right ) +C

Incorrect option

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